A

Step 1: Given Definitions
We have the tangential acceleration = a and normal acceleration n = bt^4. The total acceleration can be expressed as:
$$ a_{total} = \sqrt{a^2 + n^2} $$
Step 2: Calculate Normal Acceleration
Normal acceleration is given by:
$$ n = bt^4 \text{ (substitute this in total acceleration)} $$
Step 3: Total Acceleration
Now, substituting normal acceleration in total acceleration formula:
$$ a_{total} = \sqrt{a^2 + (bt^4)^2} $$
Step 4: Distance Relation
The distance can be related to time. Since the point starts from rest, we can find the change in velocity. Thus, for tangential acceleration, we use:
$$ v = a*t $$ and
$$ s = \frac{1}{2} a t^2 $$.
Step 5: Find Radius of Curvature
The radius of curvature R is given by:
$$ R = \frac{v^2}{n} \text{ where } v = a*t $$
Thus,
$$ R = \frac{(a t)^2}{bt^4} = \frac{a^2}{b} \cdot \frac{1}{t^2}. $$
Step 6: Conclusion
With these calculations, we show how the total acceleration and curvature radius depend on distance covered s. Therefore, both are functions of time, and ultimately of distance covered, giving us an understanding of how motion develops in terms of acceleration. Hence, the curvature radius R and total acceleration a depend on the distance s through the equations derived. Thus the answer is A.